Calculus II Video Library (Integral Calculus)

Approximating Area Under a Curve

Sigma Notation / Summation Notation Ex: Area of a Parallelogram on the Coordinate Plane Ex: Area of a Trapezoid on the Coordinate Plane Area Under a Graph Ex: Approximate Distance Traveled From a Table Using Area Ex 1: Find the Area Under a Curve Using a Geometric Formula (Rectangle) Ex 2: Find the Area Under a Curve Using a Geometric Formula (Triangle) Ex 3: Find the Area Under a Curve Using a Geometric Formula (Trapezoid) Determining Area Under Graphs Using Geometric Formulas Ex: Evaluate a Definite Integral Using Area from a Graph Ex: Definite Integration Using Geometric Formula (Line Above and Below X-Axis) Ex: Definite Integration of an Absolute Value Function Using Geometric Formula Ex: Evaluate a Definite Integral Using a Geometric Formula (Semicircle) Ex: Accumulation of Area Under a Function Using Geometric Formulas Ex: Application of Area Under a Function Using Geometric Formulas - Distance Ex 1: Reimann Sum Using a Quadratic Function (Right Endpoints and Above x-axis) Ex 2: Reimann Sum Using an Exponential Function (Left Endpoints and Above x-axis) Ex 3: Reimann Sum Using a Quadratic Function (Right Endpoints and Above/Below x-axis) Ex: Approximate the Area Under a Curve Using Rectangles (Left Using Graph) Ex: Approximate the Area Under a Curve Using Rectangles (Right Using Graph) Ex: Approximate the Area Under a Curve Using Rectangles (Midpoint Using Graph) Area Under a Graph Using Rectangles - Application Approximating Area Under a Graph Using Rectangles

Improper Integrals

Improper Integral Ex 1: Improper Integrals Ex 2: Improper Integrals Ex 3: Improper Integrals Ex 4: Improper Integrals and Area Ex: Area Using Improper Integrals Ex 1: Improper Integral - Infinite Interval (-inf,+inf) Ex 2: Improper Integral - Infinite Interval (-inf, constant) Ex 3: Improper Integral - Infinite Interval (-inf,+inf) Ex 1: Improper Integral - Discontinuous Integrand Ex 2: Improper Integral - Discontinuous Integrand Ex: Improper Integral Involving Rational Function to Find Area Under a Curve Ex: Improper Integral Involving Function with Rational Exponent to Find Area Under Curve

Introduction to Differential Equations

Introduction to Differential Equations Ex: Determine Which Functions Are Solutions to a Differential Equation Ex: Determine Which Function is a Solution to a Second Order Differential Equation Ex: Verify a Solution to a Differential Equation and Find a Particular Solution Ex: Find a Constant Function Solution to a Differential Equation Ex: Find Two Exponential Function Solutions to a Differential Equation Slope Fields Ex: Determine Which Differential Equation Would Produce a Given Direction Field Ex: Determine Direction Field Given a Solution to a Differential Equation Ex: Select a Direction Field Given a Differential Equation Using Points Differential Equations and Exponential Functions Ex: Solve a Basic Initial Value Problem (Linear)

Parametric Equations

Introduction to Parametric Equations Graphing Parametric Equations in the TI84 Converting Parametric Equation to Rectangular Form Ex 1: Write Parametric Equations as a Cartesian Equation Ex 2: Write Parametric Equations as a Cartesian Equation Ex 3: Write Parametric Equations as a Cartesian Equation Ex 4: Write Parametric Equations as a Cartesian Equation Ex: Parametric Equations Modeling a Path Around a Circle Ex: Parametric Equations for an Ellipse in Cartesian Form Ex: Find Parametric Equations For Ellipse Using Sine And Cosine From a Graph Find the Parametric Equations for a Line Segment Given an Orientation Determine Which Parametric Equations Given Would Give the Graph of the Entire Unit Circle Determine Which Parametric Equations Would Give the Graph of an Entire Line Ex 1: Find the Parametric Equations for a Lissajous Curve Ex 2: Find the Parametric Equations for a Lissajous Curve Ex 3: Find the Parametric Equations for a Lissajous Curve Ex 4: Find the Parametric Equations for a Lissajous Curve Ex: Point on a Spoke of a Rotating Wheel - Find the Radius The Derivative of Parametric Equations Ex 1: Derivatives of Parametric Equations and Applications Ex 2: Derivatives of Parametric Equations and Applications (Trig) Ex 1: Equation of a Tangent Line to a Curve Given by Parametric Equations Ex 2: Equation of a Tangent Line to a Curve Given by Parametric Equations Ex 3: Equation of a Tangent Line to a Curve Given by Parametric Equations Determine the Points Where the Tangent Lines are Horizontal or Vertical Using Parametric Equations Second Derivative of Parametric Equations: Part 1, Part 2 Ex: Determine the First and Second Derivative Given Parametric Equations First and Second Derivative of Parametric Equations -

Ex 1: Approximate the Area Under a Curve with 4 Left Sided Rectangles Ex 2: Approximate the Area Under a Curve with 4 Right Sided Rectangles Ex 3: Approximate the Area Under a Curve with 8 Left Sided Rectangles Ex 4: Approximate the Area Under a Curve with 8 Right Sided Rectangles The Antiderivative

Introduction to Antiderivatives and Indefinite Integration (No Trig) The Antiderivative Ex 1: Determine Antiderivatives Ex 2: Determine Antiderivatives Ex 3: Determine Antiderivatives Ex 4: Determine Antiderivatives Ex 5: Determine Antiderivatives Ex 1: Antiderivative Concept - Given Information about f(x), Describe F(x) Ex 2: Antiderivative Concept - Given Information about f(x), Describe F(x) Ex: Find the Particular Solution to a Basic Differential Equation Basic Antidifferentiation of Trigonometric Functions

Indefinite Integration

Ex: Basic Indefinite Integration (Polynomial, Exponential, Quotient) Ex: Indefinite Integration with a Negative Exponent Ex: Indefinite Integration Involving a Product Ex: Indefinite Integration with a Variety of Terms The Six Basic Trigonometric Integration Formulas Indefinite Integration Using Basic Trig Integral Formulas: Part 1, Part 2 Integration Involving Inverse Trig Functions: Part 1, Part 2, Part 3

Definite Integral and The Fundamental Theorem of Calculus

Ex: Solve a Basic Initial Value Problem (Exponential and Trig) Solving a differential equation by separation of variables Ex: Find the Particular Solution to a Basic Differential Equation Ex 1: Initial Value Problem Using Separation of Variables (Square Root) Ex 2: Initial Value Problem Using Separation of Variables (Square Root) Ex 1: Initial Value Problem Using Separation of Variables Involving Natural Logarithm Ex 2: Initial Value Problem Using Separation of Variables Involving Natural Logarithm Ex 1: Initial Value Problem Using Separation of Variables in the form y' = ky Ex 2: Initial Value Problem Using Separation of Variables in the Form y'=ay+b Ex: Initial Value Problem in the Form y' = kx Using Shortcut Ex: Initial Value Problem Using Separation of Variables in the Form y' = e^(ay+bx) Ex: Solve an IVP Using Separation of Variables in the Form y'=(ax+b)/(xy^2) Ex: Solve an IVP Using Separation of Variables in the Form y'=axy+bx

Applications of Integration: Business

Ex: Write a Differential Equation to Model the Change in a Bank Account Ex: Limited Growth Differential Equation Ex: Solve a Differential Equation that Models the Change in a Bank Account Balance Ex: Logistic Growth Differential Equation Ex: Complementary and Substitute Goods - Demand Function Ex: Future Value of One Time Investment Ex: Present Value of One Time Investment Given Future Value Ex 1: Future Value of Continuous Money Flow Ex 2: Continuous Money Flow needed for a Given Future Value Ex: Present Value of Continuous Money Flow

Concavity Arc Length in Parametric Form Ex 1: Determine the Arc Length of a Curve Given by Parametric Equations Ex 2: Determine the Arc Length of a Curve Given by Parametric Equations Find the Length of a Loop of a Curve Given by Parametric Equations Area Under Parametric Curves Surface Area of Revolution in Parametric Form Ex 1: Surface Area of Revolution in Parametric Form Ex 2: Surface Area of Revolution in Parametric Form

Polar Coordinates and Equations

Polar Coordinates Ex: Convert Cartesian Coordinates to Polar Coordinates Animation: Rectangular and Polar Coordinates Converting Polar Equations to Rectangular Equations Ex: Find the Rectangular and Polar Equation of a Circle From a Graph Ex: Find the Polar Equation of a Circle With Center at the Origin Ex: Find the Polar Equation for a Horizontal Line Ex: Find the Polar Equation for a Line Ex: Write a Polar Equations of a Line as a Cartesian (Rectangular) Equations Ex: Find the Polar Equation for a Parabola Ex: Find the Rectangular Equation of a Circle from a Polar Equation Ex: Convert a Polar Equation to a Rectangular Equation

Graphing Polar Equations

Graph Polar Equations I Graph Polar Equations II Animation: Graph Polar Equations Ex: Determine the Type of Conic Section Given a Polar Equation Graph Conic Sections in Polar Form: Part 1, Part 2, Part 3 Conics in Polar Form and Graphing a Parabola in Polar

The Definition of The Definite Integral The Definite Integral Ex: Setting Up a Definite Integral To Determine Area Under a Function Ex: Definite Integral as Area Given a Graph (Function) Ex: Definite Integral as Area Given a Graph (Function + Constant) Ex: Definite Integral as Area Given a Graph (Constant*Function) Ex: Evaluate Definite Integral Using Area Above and Below the x-axis The Fundamental Theorem of Calculus Proof of the Fundamental Theorem of Calculus (Part 2) Ex: Evaluate a Definite Integral on the TI-84 Ex: Graph and Evaluate a Definite Integral on the TI84

Ex: Evaluate a Basic Definite Integral of a Constant Function Using the FTC Ex: Evaluate a Basic Definite Integral of a Basic Linear Function Using the FTC Ex: Evaluate a Basic Definite Integral of a Basic Quadratic Function Using the FTC Ex: Evaluate a Basic Definite Integral of Cosine Using the FTC Ex: Fundamental Theorem of Calculus Concept Check Ex: Property of Definite Integral Subtraction Ex: Property of Definite Integral Addition Ex: Evaluate a Definite Integral of a Basic Quotient - Area Under a Curve Ex: Evaluate a Definite Integral of a Polynomial Local Maximum and Local Minimum of a Definite Integral Function (Accumulation Function) Ex 1: Area Under a Constant Function Using Definite Integration Ex 2: Area Under a Linear Function Using Definite Integration Ex 3: Area Under a Quadratic Function Using Definite Integration Ex 4: Area Under a Rational Function Using Definite Integration Ex 5: Area Under a Piece Wise Defined Function Using Definite Integration

Ex: Integration Application - Present Value for Business Ex: Future and Present Value of Continuous Money Flow Ex: Present Value of Perpetual Money Flow Ex: Determine the Present Value of a Continuous Income Stream on the TI84 (Linear) Ex: Point of Equilibrium Consumer and Producer Surplus Ex: Consumer Surplus (Linear) Ex: Producer Surplus (Linear) Ex: Consumer Surplus Ex: Producer Surplus

Applications of Integration: Volume of Revolution

Determine Volume Of Solids by Slices - Integration Application Ex 1: Volume of a Solid with Known Cross Section Using Integration - Volume by Slices Ex 2: Volume of a Solid with Known Cross Section Using Integration - Volume by Slices Ex 3: Volume of a Solid with Known Cross Section Using Integration - Volume by Slices Ex 4: Volume of a Solid with Known Cross Section Using Integration - Volume by Slices Ex 5: Volume of a Solid with Known Cross Section Using Integration - Pyramid Ex: Volume of a Solid With Slices Parallel to X-axis (Triangle) Volume of Revolution - The Disk Method Ex: Volume of Revolution - Disk Method (y=x^(1/3) Ex: Volume of Revolution - Disk Method (Quadratic Function) Ex: Volume of Revolution - Disk Method (Exponential Function base e) Ex 1: Volume of Revolution Using the Disk Method (Rational Function about y = 1) Ex 2: Volume of Revolution Using the Disk Method (Sine Squared Function) Ex 3: Volume of Revolution Using the Disk Method (Exponential Function) Volume of Revolution - The Washer Method about the x-axis Volume of Revolution - The Washer Method about the y-

Form Graphing an Ellipse in Polar Form Graphing a Hyperbola in Polar Form Ex: Find the Intercepts and Foci of a Ellipse Given a Polar Equation Ex: Find the Intercepts and Foci of a Hyperbola Given a Polar Equation Ex: Find the Intercepts and Focus of a Parabola Given a Polar Equation

Derivatives and Integrals with Polar Equations

Ex: Determine the Slope of a Tangent Line to a Polar Curve at a Given Angle Ex: Determine Where a Polar Curve Has a Horizontal Tangent Line Area using Polar Coordinates: Part 1, Part 2, Part 3 Ex: Find the Area Bounded by a Polar Curve Over a Given Interval (Spiral) Ex: Find the Area of a Inner Loop of a Limacon (Area Bounded by Polar Curve) Ex: Find the Area of Petal of a Rose (Area Bounded by Polar Curve) Area between Polar Curves: Part 1, Part 2 Ex: Find the Area of a Region Bounded by a Polar Curve (r=Acos(n*theta)) Ex 1: Find the Area of a Region Bounded by Two Polar Curves Ex 2: Find the Area of a Region Bounded by Two Polar Curves The Slope of a Tangent Line to a Polar Curve Horizontal and Vertical Tangent Lines to a Polar Curve Arc Length of a Polar Curve Ex 1: Arc Length of a Polar Curve Ex 2: Arc Length of a Polar Curve Ex: Find the Perimeter of a Region Bounded by Two Polar Curves Surface Area of Revolution of a Polar Curve

Vectors in 2D

Introduction to Vectors Vector Operations

Ex: Definite Integral Involving a Basic Linear Function Ex: Definite Integral Involving a Basic Rational Function Ex: Definite Integral Involving a Rational Function Requiring Simplifying Ex: Definite Integration Application - Cars Passing Through an Intersection Ex: Definite Integration Involving a Basic Trig Function (nonnegative) Ex: Definite Integration Involving a Basic Trig Function (above and below x-axis) Determining the value of a definite integral on the graphing calculator

The Second Fundamental Theorem of Calculus

The Second Fundamental Theorem of Calculus Proof of the Fundamental Theorem of Calculus (Part 1) Ex 1: The Second Fundamental Theorem of Calculus Ex 2: The Second Fundamental Theorem of Calculus (Reverse Order) Ex 3: The Second Fundamental Theorem of Calculus Ex 4: The Second Fundamental Theorem of Calculus with Chain Rule Ex 5: The Second Fundamental Theorem of Calculus with Chain Rule Ex 6: Second Fundamental Theorem of Calculus with Chain Rule Ex 7: Second Fundamental Theorem of Calculus with Chain Rule Ex: Evaluate a Definite Integral and the Derivative of an Integral Using a Graph

Applications of Definite Integration

Ex: Interpret the Meaning of Area Under a Function Ex 1: Application of Definite Integration (Accumulated Sales) Ex 2: Application of Definite Integration (Distance) Ex: Definite Integration Application - Velocity and Distance Ex 1: Integration Application - Work Lifting an Object Ex 2: Integration Application - Work Lifting an Object and Cable Ex: Find the Work Lifting a Leaking Bucket of Sand Given

axis Volume of Revolution - The Washer Method NOT about the x or y axis Ex 1: Volume of Revolution Using Washer Method About y = 3 Ex 2: Volume of Revolution Using Washer Method About y = 3 Ex 1: Volume of Revolution Using Washer Method About Y-Axis Ex 2: Volume of Revolution Using Washer Method About Y-Axis Ex: Volume of Revolution Using Washer Method About x=5 Volume of Revolution - The Shell Method about the x-axis Volume of Revolution - The Shell Method about the y-axis Ex: Determine a Volume of Revolution Using the Shell (tube) Method (Quadratic About y-axis)

Ex: Determine a Volume of Revolution Using the Shell

(tubes) Method (y-axis) - Calculator

Volume of Revolution - The Shell Method NOT about x or

y axis

Ex: Volume of Revolution Using the Shell Method (Basic

Quadratic about y axis)

Ex: Volume of Revolution Using the Shell Method

(Quadratic about y axis)

Ex: Volume of Revolution Using the Shell Method (Sine

about y axis)

Ex: Volume of Revolution Using the Shell Method

(Exponential about y axis)

Ex: Volume of Revolution Using Shell Method with

Horizontal Axis (Not X-Axis)

Ex: Volume of Revolution Using Shell Method with

Vertical Axis (Not Y-Axis)

Volume of Revolution - Comparing the Washer and Shell

Method

Applications of Integration: Arc Length, Surface Area, Work, Force, Center of Mass

Arc Length – Part 1 Arc Length – Part 2

Unit Vector Ex: Find the Sum of Two Vectors From a Graph (2 Dimensions) Ex: 2D Vector Scalar Multiplication Ex: Find the Unit Vector Given the Graph of a Vector in 2D Ex: Find the Difference of Two Vectors in Component Form Ex: Find the Sum of Two Vectors Given in Linear Combination Form Ex: Find the Difference of Two Vector Given in Linear Combination Form Ex: Find the Difference of Scalar Multiples of Vectors in 2D Ex: Dot Product of Vectors - 2D Ex: Dot Product of Vectors From a Graph - 2D Ex: Find a Component of a Vector So Two Vectors are Orthogonal (Dot Product) Ex 1: Find a Vector in Component Form Given an Angle and the Magnitude (30) Ex 2: Find a Vector in Component Form Given an Angle and the Magnitude (45) Ex 3: Find a Vector in Component Form Given an Angle and the Magnitude (60) Ex 4: Find a Vector in Component Form Given an Angle and the Magnitude (90) Ex 5: Find a Vector in Component Form Given an Angle and the Magnitude (180) Ex: Find the Difference of Scalar Multiples of Vectors in 2D Ex: Find the Magnitude of The Difference of Two Vectors and The Difference of Two Magnitudes Find the Component Form of a Vector from the Graph of a Vector Ex: Find the Direction and Magnitude of a Vector in Component Form Find the Component Form of a Vector Given Magnitude and Direction Ex: Write a Vector as a Combination of Two Vectors Ex: Find the Net Force of Three Vectors and the Opposite Force Ex: Find the Coordinates of a Rotated Point Using Vectors

Applications of Vectors

Mass Ex: Find the Work Lifting a Leaking Bucket of Sand and Rope Given Mass Ex: Find the Work Required to Stretch a Spring (Integration App) Ex: Find the Force Required to Stretch a Spring (Integration App) Ex: Definite Integral of Marginal Cost to find Total Cost Properties of The Definite Integral Properties of Definite Integrals and Average Value The Mean Value Theorem for Integrals Ex: Properties of Definite Integrals - Order of Integration Ex: Properties of Definite Integrals - The Difference of Two Definite Integrals Ex: Properties of Definite Integrals - Difference and Sum of Definite Integrals Ex: Properties of Definite Integrals - Determine Limits of Integration Ex: Properties of Definite Integrals - Zero Interval Ex: Integration Application - Average Value of an Investment Account Ex: Integration Application - Average Value of Temperature Function Ex 1: Average Value of a Function Ex 2: Average Value of a Trig Function Ex: Integration Application - Average Value to Determine Average Coffee Temperature Point of Equilibrium Consumer and Producer Surplus Present and Future Value: Part 1, Part 2

Area Bounded by Two Functions

Determining Area Between Two Curves - Integration Application Area Between to Graphs Ex 1: Find Area Between a Linear and Quadratic Function (respect to x) Ex 2: Find Area Between a Linear and Exponential Function (respect to x) Ex 3: Find Area Between Two Exponential Functions (respect to x) Ex 4: Find Area Between Two Quadratic Functions (respect

Ex: Find the Arc Length of a Linear Function Ex: Find the Arc Length of a Radical Function (Rational Exponent) Ex: Find the Arc Length of a Quadratic Function Surface Area of Revolution – Part 1 Surface Area of Revolution – Part 2 Ex: Surface Area of Revolution - Linear Function Ex: Surface Area of Revolution - Sine Function Ex: Find the Surface Area of Revolution of a Cubic Function About the x-axis Ex: Find the Surface Area of Revolution of a Square Root Function About the x-axis Ex: Determine the Work Required to Pump Water Out of a Circular Cylinder Ex: Find the Surface Area of Revolution of a Quadratic Function About y-axis (Respect to x) Ex: Find the Surface Area of Revolution of a Cube Root Function About y-axis (Respect to y) Ex: Determine the Work Required to Pump Water Out of Trough (Isosceles Triangle) Ex: Determine the Work Required to Pump Water Out of Trough (Quadratic Cross Section) Ex: Find the Work Lifting a Leaking Bucket of Sand Given Mass Ex: Find the Work Lifting a Leaking Bucket of Sand and Rope Given Mass Ex: Determine the Center of Mass of Three Point Masses on the Coordinate Plane Ex: Find the Centroid of a Region Consisting of Three Rectangles Ex: Find the Centroid of a Triangular Region on the Coordinate Plane Ex: Find the Centroid of a Bounded Region Involving Two Quadratic Functions Ex: Find the Centroid of a Bounded Region Involving the Sine Function Using the TI84 Ex: Find the Hydrostatic Force on a Horizontal Plate (No Calculus) Ex: Find the Hydrostatic Force on a Vertical Plate in the Shape of an Isosceles Triangle Ex: Find the Hydrostatic Force on a Dam in the Shape of a Degree 4 Polynomial

Applications of Vectors Determining the Angle Between Two Vectors Proof of the formula for the Angle Between Two Vectors Vector Projection Proof of the Vector Projection Formula Ex: Vector Projection in Two Dimensions Ex: Find the Angle of Intersection of Two Curves Using Vectors Ex: Direction and Speed of a Plane in the Wind Using Vectors Ex: Vector App: Find an Airplane Direction In The Wind To Fly Due North Vector App: Find the Direction of a Ball Thrown From a Car Ex: Vector App - Find the Resultant Vector of a 5 Direction Walk Ex: Vector App - Find the Resultant Vector of a 2 Direction Walk Vector Applications: Force and Work

Vectors in Space

Plotting Points in 3D The Equations of the Coordinate Planes in R3 Graphing a Plane Using Intercepts Ex: Determine the Distance Between a Point and a Coordinate Plane in R3 Ex: Equation of a Sphere Given the Center and Radius The Equation of a Sphere Ex: Find the Difference of Scalar Multiples of Two Vectors in 3D (Linear Combination Form) Vectors in Space Parallel Vectors Ex: Dot Product of Vectors - 3D Ex: Find the Component of a Vector so Two Vectors are Orthogonal (3D) Ex: Find the Angle Between Two Vectors in Three Dimensions Ex: Vector Projection in Three Dimensions Ex: Find the Component Form of a Vector in Space Given the Initial and Terminal Point Ex: Find a Unit Vector in the Direction of a Given Vector in 3D

to x) Ex 1: Area Bounded by Two Functions Ex 2: Area Bounded by Two Functions (2 Regions) Ex 3: Area Bounded by Two Trig Functions Ex: Determine a Function Given The Area Between Two Functions

Integration by Substitution

Indefinite Integration Using Substitution Integration by Substitution: Part 1, Part 2 Definite Integration Using Substitution Ex 1: Indefinite Integration Using Substitution Ex 2: Indefinite Integration Using Substitution Ex 3: Indefinite Integration Using Substitution Ex 4: Integration Using Substitution Ex 5: Indefinite Integration Using Substitution Ex 6: Indefinite Integration Using Substitution Ex 7: Indefinite Integration Using Substitution Ex 8: Indefinite Integration Using Substitution Involving Trig Functions Ex 9: Indefinite Integration Using Substitution Involving Trig Functions Ex: Evaluate a Indefinite Integral Using Substitution (Form e^u) Ex: Evaluate a Indefinite Integral Using Substitution (Form ae^u with Decimals) Indefinite Integration Using Substitution (Tough) Int(x^n*sqrt(x^(n-1)+c) Ex 1: Definite Integration Using Substitution - Change Limits of Integration? Ex 2: Definite Integration Using Substitution – Change Limits of Integration? Ex: Evaluate a Definite Integral Using Substitution (Form e^u) Ex: Evaluate a Definite Integral Using Substitution (Form ae^u with Decimals) Ex: Evaluate a Definite Integral Using Substitution (Form 1/u) Ex 1: Definite Integration Using Substitution Ex 2: Definite Integration Using Substitution Ex: Indefinite Integral Using Substitution Involving a Square Root

Ex: Find the Hydrostatic Force on a Semicircle Window Submerged in Water

Integration Involving Powers of Trigonometric Functions

Trig Integrals Involving Powers of Sine and Cosine: Part 1, Part 2 Trig Integrals Involving Powers of Secant and Tangent: Part 1, Part2 Ex: Integral Using Substitution with an Odd Power of Cosine Ex: Integral Using Substitution with an Odd Power of Sine Ex: Integral Using Substitution with an Odd Power of Tangent Ex: Integral Using Substitution with an Even Power of Secant Wallis’s Formula to Integrate Powers of Sine and Cosine on [0, pi/2]

Integration Using Partial Fractions

Partial Fraction Decomposition: Part 1, Part 2 Ex: Partial Fraction Decomposition - Degree 2 / Degree 3 Integration Using Partial Fraction Decomposition: Part 1, Part 2 Ex 1: Integration Using Partial Fraction Decomposition Ex 2: Integration Using Partial Fraction Decomposition and Long Division Ex: Indefinite Integral Requiring Partial Fraction Decomposition

Integration Using Trigonometric Substitution

Integration Involving Trigonometric Substitution: Part 1, Part 2, Part 3, Part 4 Ex 1: Integration Using Trigonometric Substitution Ex 2: Integration Using Trigonometric Substitution Ex 3: Integration Using Trigonometric Substitution Ex 4: Integration Using Trigonometric Substitution Ex 5: Integration Using Trigonometric Substitution Ex 6: Integration Using Trigonometric Substitution Ex: Integration Using Trigonometric Substitution and Completing the Square

Ex: Find the Magnitude of a Vector in 3D Ex: Find the Sum of Scalar Multiples of Two Vectors in 3D (Component Form Ex: Find the Difference of Scalar Multiples of Two Vectors in 3D (Linear Combination Form) Vector Cross Products Ex: Find the Cross Product of Two Vectors Ex: Find Two Unit Vectors Orthogonal to Two Given Vectors Ex 1: Properties of Cross Products - Cross Product of a Sum and Difference Ex 2: Properties of Cross Products - Cross Product of a Sum and Difference Ex: Find the Area of a Triangle Using Vectors - 3D Ex: Find the Distance Between Two Points In Space An Application of Cross Products: Torque The Triple Scalar Product: Volume of a Parallelepiped Parametric Equations of Lines in 3D

Equations of Planes and Lines in Space

Ex: Determine the Equation of a Plane Given a Point and Normal Vector The Equation of a Plane in 3D Using Vectors Graphing a Plane in 3D Ex: Determine the Point of Intersection of a Plane and a Line. Find an Equation of a Plane Containing a Line and Orthogonal to a Given Plane Find an Equation of a Plane Containing a Given Point and the Intersection of Two Planes Determine the Linear Equation of the Intersection to Two Planes Determining the Angle Between Two Planes Determining the Distance Between a Plane and a Point Determining the Distance Between a Line and a Point Ex: Find the Distance Between Two Parallel Planes Ex: Find the Equation of the Plane Containing a Given Line and a Point Using Vectors Ex: Find the Equation of a Plane Given Three Points in the Plane Using Vectors Ex: Find the Equation of a Plane Given an Orthogonal Line (Parametric) and a Point

Ex: Indefinite Integral Using Substitution Involving a Rational Function I Ex: Indefinite Integral Using Substitution Involving a Rational Function II Definite Integration Using Substitution - Int(e^(1/x^n)/x^(n+1)) Ex: Indefinite Integral Using Substitution with Exponential and Sine Ex: Definite Integration Using Substitution Involving Sine Ex: Definite Integration Using Substitution Involving Exponential and Trig Functions Ex: Indefinite Integral Involving Arcsine with Substitution Indefinite integral: (sin(x))^2- Power Reducing Substitution Indefinite Integral: (cos(2x))^2 - Power Reducing Substitution Ex 1: Trigonometric Integration - Power Reducing Formula and U-Substitution Ex 2: Trigonometric Integration - Power Reducing Formula and U-Substitution Ex: Evaluate a Indefinite Integral Integration Tables and Substitution (cot^2(a^x)) Ex: Evaluate a Indefinite Integral Integration Tables and Substitution (sin^2(x^n)) Ex: Evaluate a Indefinite Integral Using Integration Tables Integration Application: Area Using Parametric Equations – Ellipse Integration Application: Area Using Parametric Equations - Cycloid

Integration by Parts

Integration by Parts: Basics Ex: Integration by Parts - Basic Example Integration by Parts Integration by Parts: More Examples Ex 1: Integration by Parts Ex 2: Integration by Parts Ex 3: Integration by Parts Ex 4: Integration by Parts Ex 5: Integration by Parts (Trig) Ex 6: Integration by Parts Twice Ex: Integration by Parts Involving a Radical and Natural Log Ex: Integration by Parts Involving a Trig and Linear Function

Ex 1: Definite Integration Using Trigonometric Substitution Ex 2: Definite Integration Using Trigonometric Substitution Ex: Indefinite Integral in the form x^n*sqrt(a^2+x^2) Using Trigonometric Substitition Ex: Indefinite Integral in the form x^n*sqrt(a^2+x^2) Using U-Substitition Ex: Indefinite Integral in the form x^n*sqrt(a^2 - x^2) Using Trigonometric Substitition Ex: Indefinite Integral in the form x^n*sqrt(a^2 - x^2) Using U-Substitition Ex: Indefinite Integration in the Form sqrt(a^2-x^(2n))/x^(n+1) Using Trigonometric Substiution Wallis’s Formula to Integrate Powers of Sine and Cosine on [0, pi/2]

Infinite Series

Introduction to Sequences Arithmetic Sequences Geometric Sequences Ex: Find the Formula for a Geometric Sequence Given Terms Sequences on the TI84 Graphing Calculator Limits of a Sequence Ex: Limit of a Sequence Using L'Hopital's Rule (Divergent Ex: Limit of a Sequence (cos(n)/2^n) Ex: Limit of a Sequence Using L'Hopital's Rule Twice (Convergent) Ex: Limit of a Sequence Using L'Hopital's Rule (Convergent) Ex: Limit of a Sequence (Num Degree Greater) Ex 1: Limit of a Sequence (Linear/Linear) Ex 2: Limit of a Sequence (Quadratic/Quadratic) Ex: Limit of a Sequence (Num Degree Less) The Squeeze Theorem Arithmetic Series Geometric Series Introduction to Infinite Series Infinite Series: The Nth Term Divergent Test

Ex: Find the Equation of Plane Containing a Line and Orthogonal to a Given Plane Ex: Find the Equation of a Plane Given a Point in the Plane and a Parallel Plane Ex: Find the Parametric Equations of a Line in Space Given Two Points on the Line Ex: Find the Point Where a Line in 3D Intersects the xz-plane Ex 1: Find the Parametric Equations of the Line of Intersection of Two Planes Using Vectors Ex 2: Find the Parametric Equations of the Line of Intersection of Two Planes Using Vectors Ex: Find the Parametric Equations of a Line Perpendicular to a Plane Through a Point Ex: Find Parametric Equations of a Line in Space Parallel to a Vector Containing a Given Point

Quadric Surfaces, Cylindrical Coordinates and Spherical Coordinates

Cylindrical Surfaces Introduction to Quadric Surfaces The Ellipsoid The Hyperboloid of One Sheet The Hyperboloid of Two Sheets The Elliptical Cone The Elliptical Paraboloid The Hyperbolic Paraboloid Surfaces of Revolution Cylindrical Coordinates Converting Between Cylindrical and Rectangular Equations Spherical Coordinates Converting Between Spherical and Rectangular Equations Ex 1: Convert Cartesian Coordinates to Spherical Coordinates Ex 2: Convert Cartesian Coordinates to Spherical Coordinates Ex 1: Convert Spherical Coordinates to Cartesian Coordinates Ex 2: Convert Spherical Coordinates to Cartesian Coordinates

(x*cos(4x)) Ex: Integration by Parts - Definite Integral Involving a Quadratic and Natural Log Function Ex: Definite Integral Using Integration by Parts in the Form x^n*ln(x) Ex: Definite Integral Using Integration by Parts in the Form x^(n)*ln(bx) Ex: Evaluate a Indefinite Integral Using Integration by Parts - Int(ln(ax+b),x) Ex: Integration by Parts Twice Application Ex: Integration by Parts Twice and Solving

Integration Involving Inverse Trig Function and Integration Tables

Ex: Integration Tables - Basic Integration Involving sqrt(a^2-u^2) Ex: Integration Tables - Basic Integration Involving a^2+u^2 Ex: Integration Tables - Basic Integration Involving a^2-u^2 Ex: Integration Tables - Integration Involving e^(ax)*sin(bx) Ex: Integration Table - Integration Involving 1/u and a^2+u^2 Ex: Integration Tables - Integration Requiring U-Substitution sqrt(a^2-u^2) Ex: Integration Tables - Integration Requiring U-substitution Involving sqrt(u^2+a^2) Ex: Integration Tables - Integration Involving Requiring U-substitution Involving (tan(u))^n Ex: Indefinite Integration Using U-Substitution Involving an Inverse Trig Function Ex: Definite Integration Involving Inverse Tangent - 1/sqrt(a^2-u^2) Ex: Definite Integration Involving Inverse Tangent - 1/(a^2+u^2) Ex: Definite Integration Involving Inverse Tangent with U-Substitution - 1/(a^2+u^2) Ex: Indefinite Integration Involving Arctangent Requiring U-sub and Completing the Square Ex: Indefinite Integration Involving Arctangent Requiring Completing the Square

Infinite Geometric Series Sequences and Series on the TI84 Graph Partial Sums of an Infinite Series on the TI84 Telescoping Series Ex 1: Telescoping Series (Convergent) Ex 2: Telescoping Series (Divergent) Ex 3: Telescoping Series with Partial Fractions The Harmonic Series The Integral Test Infinite Series: The Integral Test Ex: Infinite Series - Integral Test (Rational Function and Convergent) Ex: Infinite Series - Integral Test (Rational Function and Divergent) Ex: Infinite Series - Integral Test (Radical and Divergent) Ex: Infinite Series - Integral Test (Exponential and Convergent) Ex: Infinite Series - Integral Test (Convergent Involving Arctangent) Ex: Infinite Series - Integral Test Requiring Integration by Parts (Convergent) The p-series Test Infinite Series: The p-Series Test Ex 1: Infinite Series - P Series Test (Convergent) and Find a Partial Sum Ex 2: Infinite Series - P Series Test (Divergent) and Find Partial Sum The Direct Comparison Test Infinite Series: The Direct Comparison Test Ex: Infinite Series - Direct Comparison Test (Convergent) Ex: Infinite Series - Direct Comparison Test (Divergent) Ex: Infinite Series - Direct Comparison Test (Inconclusive) The Limit Comparison Test Ex: Infinite Series - Limit Comparison Test (Convergent) Ex: Infinite Series - Limit Comparison Test (Geometric, Divergent) Ex: Infinite Series - Limit Comparison Test (Radical, Convergent) Ex: Infinite Series - Limit Comparison Test (Divergent) Ex: Infinite Series - Limit Comparison Test (Radical, Divergent) Infinite Series: The Limit Comparison Test (Divergent) Infinite Series: The Limit Comparison and Direct

Ex 1: Convert Cartesian Coordinates to Cylindrical Coordinates Ex 2: Convert Cartesian Coordinates to Cylindrical Coordinates Ex: Convert Cylindrical Coordinates to Cartesian Coordinates

Vector Valued Functions

Introduction to Vector Valued Functions The Domain of a Vector Valued Function Ex: Determine the Domain of a Vector Valued Function Ex: Find the Point of Intersection of a Line Given by a Vector Function and a Coordinate Plane Ex 1: Vector Valued Function - Curve of Intersection Ex 2: Vector Valued Function - Curve of Intersection Determining a Vector Valued Function from a Rectangular Equation Determine a Vector Valued Function from the Intersection of Two Surfaces Limits of Vector Valued Functions The Derivative of a Vector Valued Function Ex: Find a Tangent Vector of a Space Curve Given by a Vector Valued Function Ex: Find the Velocity and Acceleration Vector Given the Position Vector Valued Function Ex: Find the Velocity and Position Vector Functions Given the Acceleration Vector Function Ex: Find Parametric Equations of a Tangent Line to a Space Curve Ex: Find a Unit Tangent Vector to a Space Curve Given by a Vector Valued Function Determining Where a Space Curve is Smooth from a Vector Valued Function Indefinite Integration of Vector Valued Functions Indefinite Integration of Vector Valued Functions with Initial Conditions Definite Integration of Vector Valued Functions Ex: Integrate a Vector Valued Function Properties of the Derivatives of Vector Valued Functions The Derivative of the Cross Product of Two Vector Valued Functions Determining Velocity, Speed, and Acceleration Using a

Numerical Integration

Ex 1: Numerical Integration - The Midpoint Rule Ex 2: Numerical Integration - The Midpoint Rule (Fractions) Trapezoidal Rule of Numerical Integration Ex: Numerical Integration - The Trapezoid Rule Trapezoid Rule Error - Numerical Integration Approximation Trapezoid Rule - Determine n for a Given Accuracy Simpson’s Rule of Numerical Integration Ex: Simpson's Rule Using a Table of Values Ex 1: Numerical Integration - Simpson's Rule Ex 2: Estimate a Definite Integral Using Simpson's Rule (fractional subintervals) Simpson's Rule Error - Numerical Integration Approximation Simpson's Rule - Determine n for a Given Accuracy

Comparison Tests Infinite Series: The Limit Comparison and Ratio Tests - Part 1 Infinite Series: The Limit Comparison and Ratio Tests - Part 2 The Root Test Infinite Series: The Root Test I Infinite Series: The Root Test II The Ratio Test Ex 1: Infinite Series - The Root Test (Convergent) Ex 2: Infinite Series - The Root Test (Divergent) Ex 3: Infinite Series - The Root Test (Divergent) Ex 4: Infinite Series - The Root Test (Convergent) Ex 5: Infinite Series - The Root Test (Divergent) Infinite Series: The Ratio Test I Infinite Series: The Ratio Test II Ex 1: Infinite Series - The Ratio Test (Convergent) Ex 2: Infinite Series - The Ratio Test (Divergent) Ex 3: Infinite Series - The Ratio Test (Convergent) Ex 4: Infinite Series - The Ratio Test (Convergent) The Alternating Series Test Ex: Find a Partial Sum of a Alternating Series (Method #1) Ex: Find a Partial Sum of a Alternating Series (Method #2) Conditionally and Absolutely Convergent Series Ex 1: Determine if a Series Is Conditionally Convergent, Absolutely Convergent, or Divergent Ex 2: Determine if a Series Is Conditionally Convergent, Absolutely Convergent, or Divergent Ex 3: Determine if a Series Is Conditionally Convergent, Absolutely Convergent, or Divergent Ex 4: Determine if a Series Is Conditionally Convergent, Absolutely Convergent, or Divergent Infinite Series: The Alternating Series Test Ex: Apply Alternating Series to Infinite Series - Divergent Ex: Determine if an Infinite Alternating Series Converges or Diverges (Convergent) Ex: Determine if an Infinite Alternating Series Converges or Diverges (Divergent) Ex 1: Determine if an Series and an Alternating Series Converge or Diverge Ex 2: Determine if an Series and an Alternating Series Converge or Diverge Ex: Find the Error When Using a Partial Sum to Estimate

Vector Valued Function Determining the Unit Tangent Vector Determining the Unit Normal Vector Proving the Unit Normal Vector Formula Determining a Tangent Line of a Curve Defined by a Vector Valued Function Determining the Tangential and Normal Components of Acceleration Determining Arc Length of a Curve Defined by a Vector Valued Function Ex: Determine Arc Length of a Spiral Given by Parametric Equations Ex: Determine Arc Length of a Helix Given by a Vector Valued Function Determining Curvature of a Curve Defined by a Vector Valued Function Ex 1: Find the Curvature of a Space Curve Given by a Vector Function (Cross Product) Ex 2A: Find the Curvature of a Space Curve Given by a Vector Function (Cross Product) Ex 2B: Find the Curvature of a Space Curve Given by a Vector Function (No Cross Product) Find the Angle of Intersection of Two Space Curves Given As Vector Functions Determining the Binormal Vector

Proofs

Proof of the Fundamental Theorem of Calculus (Part 1) Proof of the Fundamental Theorem of Calculus (Part 2) Proof of the formula for the Angle Between Two Vectors Vector Projection Proof of the Vector Projection Formula

an Infinite Sum (Alternating Series) Ex: Number of Terms Needed in Partial Sum to Estimate an Infinite Sum with a Given Error Taylor Polynomials Taylor’s Theorem with Remainder

Power Series

Power Series: Part 1, Part 2 Representing a Function as a Geometric Power Series: Part 1, Part 2 Ex 1: Interval of Convergence for Power Series (Centered at 0) Ex 2: Interval of Convergence for Power Series (Centered at 0) Ex 3: Interval of Convergence for Power Series (Centered at 0) Ex 4: Interval of Convergence for Power Series (Centered at 0) Ex 5: Interval of Convergence for Power Series (Not Centered at 0) Ex 6: Interval of Convergence for Power Series (Not Centered at 0) Taylor and Maclaurin Series Using Power Series Tables – Part 1, Part 2 Ex 1: Maclaurin Series and Polynomial of cos(2x) / Find Approximation Error Ex: Find the Taylor Series of x^3 Ex: Find the Taylor Series of e^x Ex: Find a Degree One and Degree Two Maclaurin Polynomial Determine the Maclaurin Series and Polynomial for Function in the Form a*cos(bx^2) Determine the Maclaurin Series and Polynomial for Function in the Form ax^2*e^(bx) Determine the Maclaurin Series and Polynomial for Function in the Form ax^2*sin(bx) Ex: Determine a Taylor Polynomial for a Square Root Function Ex: Find a Maclaurin Polynomial and Error of an Approximation - ln(cos(x)) Ex: Find a Maclaurin Polynomial and the Interval for a Given Error - cos(x)

Ex: Find a Maclaurin Polynomial and the Interval for a Given Error - ln(1+x) Ex: Use a Maclaurin Polynomial for sin(bx^n) to Approximate an Integral Ex 1: Find a Power Series to Represent a Rational Function Ex 2: Find a Power Series to Represent a Rational Function Ex 3: Find a Power Series to Represent a Power Series Ex: Find a Power Series to Represent a Power Series Using a Product Ex: Find a Power Series to Represent a arctan(x) Using Integration Ex: Find a Power Series to Represent a Rational Function Using Differentiation Differentiating and Integrating Using Power Series Ex: Determine a Simplified Power Series for a Function Involving e^(ax) Ex 1: Determine the Sum of a Power Series Ex 2: Determine the Sum of a Power Series Estimate a Definite Integral using a Power Series (Rational Function)